Finds the side length and area of the largest square that fits entirely inside a circle of a given radius.
How it works
The inscribed square’s diagonal equals the circle’s diameter, so the side length is the radius multiplied by the square root of 2.
What this does not include
This does not include the reverse problem (the largest circle that fits inside a square) — that uses a different relationship entirely.
How to use this calculator
- Enter the circle’s radius.
A worked example
A circle with radius 5, largest inscribed square: side = radius × √2 = 7.0711, area = 50.
Radius 10: side = 14.1421, area = 200.
What the variables mean
| Variable | Meaning |
|---|---|
| Radius | Radius of the circle the square is inscribed within |
Edge cases worth knowing
The square’s diagonal exactly equals the circle’s diameter — the geometric relationship this formula is built on, since the largest square that fits must have all four corners touching the circle.
A radius of zero collapses both the circle and the square to a point, so the calculator declines to show a result.
Frequently asked questions
Why does the square’s diagonal equal the circle’s diameter?
Because the largest possible square touches the circle at all four corners, and the longest line you can draw across that square (its diagonal) is exactly the circle’s widest point.
What fraction of the circle does the square fill?
About 63.7% of the circle’s area — the square always fills the same proportion regardless of the circle’s size.
What’s a real-world use for this calculation?
Cutting the largest possible square panel from a circular piece of material, like glass or fabric.
